Complete $(2+1)$-dimensional Ricci flow spacetimes

TL;DR

This paper proves that in 2+1 dimensions, complete and regular Ricci flow spacetimes are necessarily cylindrical, meaning they are equivalent to classical Ricci flows on a fixed surface, confirming a rigidity property.

Contribution

It establishes a rigidity result for complete 2+1-dimensional Ricci flow spacetimes, showing they must be cylindrical and equivalent to classical Ricci flows.

Findings

Complete 2+1-dimensional Ricci flow spacetimes are cylindrical.

Such spacetimes are isometric to classical Ricci flows on a fixed surface.

The result confirms a rigidity property in this setting.

Abstract

Ricci flow spacetimes were introduced by Kleiner & Lott as a way to describe Ricci flow through singularities, and have since been used elsewhere in the literature, prompting the question of their rigidity. In $(2+1)$-dimensions, we show that every complete and sufficiently regular spacetime must be a cylindrical spacetime. That is, if the metric is complete on each spatial slice, after imposing a necessary continuity condition, we can conclude that every spatial slice must be diffeomorphic to a fixed surface, and the Ricci flow spacetime is isometric to a classical Ricci flow on this surface.